Am i using rivalry corretly in the sentences and do they make sense?

[scott833]

1) There is a rivalry between these two teams.
2) The rivalry between two team had started since a long time ago and is still ongoing.
3) The derby football matches refer to football matches between two rival teams from the same city or state.
4) Two teams from the same city always compete with and dislike each other and it is called the rivalry between two local teams.

* I just watched the champions league final, i am trying to improve my vocabulary and English with regard to football.

 

[emsr2d2]

1) There is a rivalry between these two teams. Correct.
2) The rivalry between the two teams [STRIKE]had[/STRIKE] started since a long time ago and is still ongoing. Use of rivalry correct but see other amendments.
3) [STRIKE]The[/STRIKE] "Derby" football matches refer to football matches between two rival teams from the same city or state. Again, use of rival is correct but see amendment.
4) Two teams from the same city always compete with and dislike each other. [STRIKE]and[/STRIKE] It is called [STRIKE]the[/STRIKE] rivalry [between two local teams].
You don't need the article here and you don't really need to specify "between two local teams" as that is already implicit in the opening sentence. This works better broken into two sentences.

* I just watched the champions league final, i am trying to improve my vocabulary and English with regard to football.

Please see amendments and comments above.

 

[SoothingDave]

The rivalry between the two teams started since a long time ago and is still ongoing.

I think you meant to scratch out the word "since" as well. The rivalry started a long time ago...